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Degrees of freedom (d.f.) in statistics represent the number of independent values or quantities which can be assigned to a statistical distribution. In the context of variance, they play a crucial role in accurately estimating the population parameter from a sample statistic.

In the exercise, the degrees of freedom for the pooled variance estimator are calculated using the formula: \( d.f. = n_1 + n_2 - 2 \), where \( n_1 \) and \( n_2 \) are the sample sizes of two groups. The subtraction of 2 accounts for the fact that we have estimated two parameters (in this case, the variances of the two samples) from the data. This adjustment is necessary to ensure an unbiased estimate of the population variance.

Degrees of freedom also affect the shape of the t-distribution, which is used when working with small samples. As the degrees of freedom increase, the t-distribution approaches the normal distribution. In the provided exercise, the degrees of freedom were determined to be 12, which has implications for hypothesis testing and confidence intervals involving the pooled variance.

In statistics, a variance estimator is a formula used to calculate the measure of dispersion within a data set. The pooled variance estimator combines variances from two independent samples to obtain a single, more accurate estimate of the population variance, especially when the population variances are assumed to be equal.

The formula for the pooled variance \( s^2 \) is: \[ s^2 = \frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2} \]. This formula weighs the sample variances \( s_1^2 \) and \( s_2^2 \) by their respective degrees of freedom, \( n_1-1 \) and \( n_2-1 \). The result is an estimator that takes into account the sample sizes and variability of each sample.

Understanding how to compute the pooled variance is crucial when performing F-tests or ANOVA (Analysis of Variance) tests, where the homogeneity of variances is a key assumption. Our exercise demonstrates the computation step-by-step, yielding a pooled variance of 3.775. This pooled estimate is more robust than individual sample variances for subsequent inferential statistics.

Statistics formulas are the backbone of data analysis, providing the methods to summarize, describe, and make inferences about the collected data. Formulas such as the one for pooled variance allow us to condense complex data into comprehensible figures that can be used for decision making.

The application of the appropriate formula, however, must be done with understanding of the underlying assumptions and context. When using the pooled variance formula, it's important to be aware that we are assuming the populations from which the samples are taken have the same variance. Without this assumption, a different approach would be needed.

Formulas for calculating other important statistics include the mean, standard deviation, and correlation coefficient. Each of these requires knowledge of the correct degrees of freedom to ensure the validity of the results. In essence, statistics formulas turn raw data into actionable insights, and mastering their use is a fundamental skill for any student of statistics.